The layer number of $\alpha$-evenly distributed point sets

TL;DR

This paper studies the layer number in peeling processes for $ ext{alpha}$-evenly distributed point sets, providing new bounds that extend known results for random and grid point sets to more general distributions in various dimensions.

Contribution

It introduces bounds on the layer number for $ ext{alpha}$-evenly distributed point sets, generalizing previous results for random and grid sets to higher dimensions and broader distributions.

Findings

Upper bound of $O(|X|^{3/4})$ in the plane for $ ext{alpha}$-evenly distributed sets

Upper bound of $O(|X|^{(d+1)/2d})$ in $ ext{R}^d$ for $d extgreater 2$

Explicit construction showing the bounds are tight or cannot be improved.

Abstract

For a finite point set in $\mathbb{R}^d$, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number $L(X)$ of a given point set $X$ is defined as the number of steps of the peeling process in order to delete all points in $X$. It is known that if $X$ is a set of random points in $\mathbb{R}^d$, then the expectation of $L(X)$ is $\Theta(|X|^{2/(d+1)})$, and recently it was shown that if $X$ is a point set of the square grid on the plane, then $L(X)=\Theta(|X|^{2/3})$. In this paper, we investigate the layer number of $\alpha$-evenly distributed point sets for $\alpha>1$; these point sets share the regularity aspect of random point sets but in a more general setting. The set of lattice points is also an $\alpha$-evenly distributed point set for some $\alpha>1$. We find an upper bound of $O(|X|^{3/4})$ for the layer number of an $\alpha$-evenly distributed point set $X$ in a unit disk on the plane for some $\alpha>1$, and provide an explicit construction that shows the growth rate of this upper bound cannot be improved. In addition, we give an upper bound of $O(|X|^{\frac{d+1}{2d}})$ for the layer number of an $\alpha$-evenly distributed point set $X$ in a unit ball in $\mathbb{R}^d$ for some $\alpha>1$ and $d\geq 3$.